Permanence and Global Attractivity of a Discrete Two-Prey One-Predator Model with Infinite Delay

Abstract

A discrete two-prey one-predator model with infinite delay is proposed. A set of sufficient conditions which guarantee the permanence of the system is obtained. By constructing a suitable Lyapunov functional, we also obtain sufficient conditions ensuring the global attractivity of the system. An example together with its numerical simulation shows the feasibility of the main results.

1. Introduction

The aim of this paper is to investigate the persistence and stability property of the following discrete two-prey one-predator model with infinite delays:

𝑥₁(𝑛+1)=𝑥₁(𝑛)exp[𝑎₁(𝑛)−𝑏₁(𝑛)𝑥₁(𝑛)−𝑐₁(𝑛)^𝑛_𝑠=−∞𝐻₁(𝑛−𝑠)𝑥₃𝑥(𝑠)],₂(𝑛+1)=𝑥₂(𝑛)exp[𝑎₂(𝑛)−𝑏₂(𝑛)𝑥₂(𝑛)−𝑐₂(𝑛)^𝑛_𝑠=−∞𝐻₂(𝑛−𝑠)𝑥₃𝑥(𝑠)],₃(𝑛+1)=𝑥₃(𝑛)exp[−𝑏₃(𝑛)𝑥₃(𝑛)+𝑑₁(𝑛)^𝑛_𝑠=−∞𝐻₃(𝑛−𝑠)𝑥₁(𝑠)+𝑑₂(𝑛)^𝑛_𝑠=−∞𝐻₄(𝑛−𝑠)𝑥₂(𝑠)], (1.1) where 𝑥_𝑖(𝑛), 𝑖=1,2, are the densities of the prey species 𝑖 at the 𝑛th generation; 𝑥₃(𝑛) is the density of the predator at the 𝑛th generation; {𝑎_𝑖(𝑛)}, {𝑐_𝑖(𝑛)}, {𝑑_𝑖(𝑛)}(𝑖=1,2), {𝑏_𝑖(𝑛)}(𝑖=1,2,3); {𝐻_𝑖(𝑛)}(𝑖=1,…,4) are all bounded nonnegative sequences such that

0<𝑎^𝑙_𝑖≤𝑎^𝑢_𝑖,0<𝑐^𝑙_𝑖≤𝑐^𝑢_𝑖,0<𝑑^𝑙_𝑖≤𝑑^𝑢_𝑖,0<𝑏^𝑙_𝑖≤𝑏^𝑢_𝑖,^∞_𝑛=0𝐻_𝑖(𝑛)=1. (1.2) Here, for any bounded sequence {𝑔(𝑛)}, set 𝑔^𝑢=sup_𝑛∈𝑁{𝑔(𝑛)} and 𝑔^𝑙=inf_𝑛∈𝑁{𝑔(𝑛)}.

From the point of view of biology, in the sequel, we assume that

𝑥_𝑖(𝑠)=Φ_𝑖(𝑠)≥0,Φ_𝑖(0)>0,𝑖=1,2,3, (1.3) where 𝑠=…,−𝑛,−𝑛+1,…,−1,0. Then system (1.1) with the initial condition (1.3) has a unique positive solution (𝑥₁(𝑛),𝑥₂(𝑛),𝑥₃(𝑛))^𝑇.

As one of the dominant themes in mathematical biology, the predator-prey relationship has been studied in a number of ways (see [1–4] and the references therein). In 1970, Parrish and Saila [5] firstly proposed the one-prey two-predator model as follows:

𝑥̇₁(𝑡)=𝑥₁(𝑡)(𝑏₁−𝑥₁(𝑡)−𝛼𝑥₂𝑥̇(𝑡)−𝜂𝑧(𝑡)),₂(𝑡)=𝑥₂(𝑡)(𝑏₂−𝛽𝑥₁(𝑡)−𝑥₂̇(𝑡)−𝜐𝑧(𝑡)),𝑧(𝑡)=𝑧(𝑡)(−𝑏₃+𝑑𝜂𝑥₁(𝑡)+𝑑𝜐𝑥₂(𝑡)). (1.4) Gramer and May [6] studied the stability of the positive equilibrium of system (1.4); Takcuchi and Adachi [7] investigated the existence of the positive equilibrium and Hopf Bifurcation of the above system.

Recently, Elettreby [8] proposed the following two-prey one-predator model:

̇̇̇𝑥(𝑡)=𝑎𝑥(𝑡)(1−𝑥(𝑡))−(𝑥(𝑡)𝑧(𝑡)),𝑦(𝑡)=𝑏𝑦(𝑡)(1−𝑦(𝑡))−(𝑦(𝑡)𝑧(𝑡)),𝑧(𝑡)=−𝑐𝑧(𝑡)²+𝑑𝑥(𝑡)𝑧(𝑡)+𝑒𝑦(𝑡)𝑧(𝑡), (1.5) where all the parameters in system (1.5) are positive constants. By applying differential inequality theory and iterative scheme, he showed that the unique positive equilibrium of system (1.5) is globally attractive. It is well known that a suitable ecosystem should incorporate some pase of the state of system, which is represented by time delays. Li et al. [9] studied the two-prey one-predator model with delays:

𝑥̇₁(𝑡)=𝑥₁(𝑡)(𝑏₁−𝑥₁(𝑡)−𝛼𝑥₂(𝑡)−𝑟₁𝑥̇𝑦(𝑡)),₂(𝑡)=𝑥₂(𝑡)(𝑏₂−𝛽𝑥₁(𝑡)−𝑥₂(𝑡)−𝑟₂̇𝑦(𝑡)),𝑦(𝑡)=−𝑏₃𝑦(𝑡)+𝑑𝑟₁𝑥₁(𝑡−𝜏)𝑦(𝑡−𝜏)+𝑑𝑟₂𝑥₂(𝑡−𝜏)𝑦(𝑡−𝜏), (1.6) where 𝑥₁(𝑡), 𝑥₂(𝑡), 𝑦(𝑡) are the densities of the prey and predator at the time 𝑡, respectively, 𝑏₁, 𝑏₂, 𝑏₃, 𝛼, 𝛽, 𝑟₁, 𝑟₂, 𝑑 are all positive constants. They investigated the Hopf bifurcation of system (1.6). Corresponding to system (1.5), Huang [10] proposed and studied the following system with infinite delays:

𝑥̇₁(𝑡)=𝑎𝑥₁(𝑡)(1−𝑥₁1(𝑡)−𝑎^𝑡_−∞𝐾₁(𝑡−𝑠)𝑥₃𝑥̇(𝑠)𝑑𝑠),₂(𝑡)=𝑏𝑥₂(𝑡)(1−𝑥₂1(𝑡)−𝑏^𝑡_−∞𝐾₂(𝑡−𝑠)𝑥₃𝑥̇(𝑠)𝑑𝑠),₃(𝑡)=𝑥₃(𝑡)(−𝑐𝑥₃(𝑡)+𝑑^𝑡_−∞𝐾₃(𝑡−𝑠)𝑥₁(𝑠)𝑑𝑠+𝑒^𝑡_−∞𝐾₄(𝑡−𝑠)𝑥₂(𝑠)𝑑𝑠), (1.7) where all the coefficients 𝑎, 𝑏, 𝑐, 𝑑, 𝑒 are positive constants, and 𝐾_𝑖∶[0,+∞)→(0,+∞) are continue functions such that ∫^∞₀𝐾_𝑖(𝑠)𝑑𝑠=1,𝑖=1,2,3,4. By applying iterative scheme, he showed that the unique positive equilibrium of the system is globally attractive.

On the other hand, it is well known that the discrete time model governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Corresponding to traditional continuous Logistic model governed by differential equations, Mohamad and Gopalsamy [11] proposed the following single species discrete model:

𝑥(𝑛+1)=𝑥(𝑛)exp[𝑟(𝑛)(1−𝑥(𝑛)𝐾(𝑛))]. (1.8) They tried to obtain a set of sufficient conditions which ensure that (1.8) admits a unique positive and globally asymptotically stable almost periodic solution. However, Zhou and Zou [12] gave an counterexample which shows that the main results of [11] are not correct. By developing some new analysis technique, Zhou and Zou [12] obtained sufficient conditions which ensure the existence of a positive and globally asymptotically stable 𝜔-periodic solution of system (1.8). Chen and Zhou [13] further generalized system (1.8) to the following two-species Lotka-Volterra competition system:

𝑥(𝑛+1)=𝑥(𝑛)exp[𝑟₁(𝑛)(1−𝑥(𝑛)𝐾₁(𝑛)−𝜇₂(𝑛)𝑦(𝑛))],𝑦(𝑛+1)=𝑦(𝑛)exp[𝑟₂(𝑛)(1−𝜇₁(𝑛)𝑥(𝑛)−𝑦(𝑛)𝐾₂(𝑛))]. (1.9) They obtained the sufficient conditions which guarantee the persistence of the system (1.3). Also, for the periodic case, they obtained the sufficient conditions which guarantee the existence of a globally stable periodic solution of the system. Wang and Lu [14] proposed the following Lotka-Volterra model:

𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp[𝑟_𝑖(𝑘)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑘)𝑥_𝑗(𝑘)],𝑖=1,2,…,𝑛, (1.10) where 𝑥_𝑖(𝑘) is the density of population 𝑖 at 𝑘th generation; 𝑟_𝑖(𝑘) is the growth rate of population 𝑖 at 𝑘th generation; 𝑎_𝑖𝑗(𝑘) measures the intensity of intraspecific competition or interspecific action of species. By constructing a suitable Lyapunov function and using the finite covering theorem of Mathematic Analysis, they obtained a set of sufficient conditions which ensures the system to be globally asymptotically stable. Similar to the continuous ones, some scholars also argued that the discrete model should incorporate some past state of the species and thus should consider the discrete model with time delay. Recently, Chen [15] investigated the persistent property of the following discrete two species Lotka-Volterra competition model with deviating arguments:

𝑥₁(𝑛+1)=𝑥₁(𝑛)exp[𝑟₁𝑥(𝑛)(1−¹(𝑛)𝐾₁(𝑛)−𝜇₂(𝑛)^𝑛_𝑠=−∞𝐻₁(𝑛−𝑠)𝑥₂𝑥(𝑠))],₂(𝑛+1)=𝑥₂(𝑛)exp[𝑟₂(𝑛)(1−𝜇₁(𝑛)^𝑛_𝑠=−∞𝐻₂(𝑛−𝑠)𝑥₁𝑥(𝑠)−²(𝑛)𝐾₂(𝑛))], (1.11) where 𝑥_𝑖(𝑛), 𝑖=1,2, are the densities of competition species 𝑖 at 𝑛th generation. By establishing a new difference inequality, Chen [15] showed that under the same conditions as that of Chen and Zhou [13], (1.11) is also permanent, which means that with some suitable restriction on the coefficients of the system, delay has no influence on the persistent property of the system. Chen [16] also investigated the persistent property of a discrete 𝑛-species nonautonomous Lotka-Volterra competitive systems with infinite delays and feedback controls. As we can see, both [15] and [16] considered the persistent property of the system, but they did not investigate the stability property of the system. Recently, Chen et al. [17] investigated the dynamic behaviors of the following general discrete nonautonomous system of plankton allelopathy with finite time delay:

𝑁₁(𝑘+1)=𝑁₁(𝑘)exp[𝑟₁(𝑘)−^𝑚_𝑙=0𝑎_1𝑙(𝑘)𝑁₁(𝑘−𝑙)−^𝑚_𝑙=0𝑏_1𝑙(𝑘)𝑁₂−(𝑘−𝑙)_𝑚𝑙=0𝑐_1𝑙(𝑘)𝑁₁(𝑘)𝑁₂𝑁(𝑘−𝑙)],₂(𝑘+1)=𝑁₂(𝑘)exp[𝑟₂(𝑘)−^𝑚_𝑙=0𝑎_2𝑙(𝑘)𝑁₂(𝑘−𝑙)−^𝑚_𝑙=0𝑏_2𝑙(𝑘)𝑁₁−(𝑘−𝑙)_𝑚𝑙=0𝑐_2𝑙(𝑘)𝑁₂(𝑘)𝑁₁(𝑘−𝑙)], (1.12) where 𝑁_𝑖(𝑘) represents the densities of population 𝑖 at the 𝑘th generation; 𝑟_𝑖(𝑘) is the intrinsic growth rate of population 𝑖 at the 𝑘th generation; 𝑎_𝑖𝑙(𝑘) measures the intraspecific influence of the (𝑘−𝑙)th generation of population 𝑖 on the density of own population; 𝑏_𝑖𝑙(𝑘) stands for the inter-specific influence of the (𝑘−𝑙)th generation of population 𝑖 on the density of own population and 𝑐_𝑖𝑙(𝑘) stands for the effect of toxic inhibition of population 𝑖 by population 𝑗 at the (𝑘−𝑙)th generation, 𝑖,𝑗=1,2 and 𝑖≠𝑗. {𝑟_𝑖(𝑘)}, {𝑎_𝑖𝑙(𝑘)}, {𝑏_𝑖𝑙(𝑘)} and {𝑐_𝑖𝑙(𝑘)} are all bounded nonnegative sequences defined for 𝑘∈𝑁. They obtained sufficient conditions which guarantee the permanence, global attractivity and partial extinction of the above system.

Concerned with the discrete predator-prey system, by giving the detail analysis of the right hand side of the system, Yang [18] obtained a set of sufficient conditions which ensures the uniform persistence of the system investigated. Recently, Chen and Chen [19] proposed the following discrete periodic Volterra model with mutual interference and Holling II type functional response

𝑥(𝑛+1)=𝑥(𝑛)exp[𝑟₁(𝑛)−𝑏₁𝑐(𝑛)𝑥(𝑛)−¹(𝑛)𝑦𝑘+𝑥(𝑛)^𝑚(𝑛)],𝑦(𝑛+1)=𝑦(𝑛)exp[−𝑟₂(𝑛)−𝑏₂𝑐(𝑛)𝑦(𝑛)−²(𝑛)𝑥(𝑛)𝑦𝑘+𝑥(𝑛)^𝑚−1(𝑛)]. (1.13) They also obtained sufficient conditions which ensure the permanence of the system. For more works on discrete population model, one could refer to [11–42] and the references cited therein.

However, to the best of the authors knowledge, to this day, no scholars propose and study the discrete predator-prey model with infinite delays. This motivates us to propose and study (1.1). The aim of this paper is to investigate the persistent and stability property of system (1.1).

The rest of the paper is arranged as follows: some useful lemmas are stated in the following section. Sufficient conditions which ensure the permanence and global attractivity of system (1.1) are stated and proved in Section 3. In Section 4, an example together with its numeric simulation shows the feasibility of the main results. We end this paper by a brief discussion.

2. Preliminaries

Now let us state several lemmas which will be useful in proving main results.

Lemma 2.1 (see [29]). Assume that {𝑥(𝑛)} satisfies 𝑥(𝑛)>0 and 𝑥(𝑛+1)≤𝑥(𝑛)exp[𝑟(𝑛)−𝑎(𝑛)𝑥(𝑛)], (2.1) for 𝑛∈𝑁, where {𝑟(𝑛)} and {𝑎(𝑛)} are all positive sequences bounded above and below by positive constants. Then limsup_𝑛→+∞1𝑥(𝑛)≤𝑎^𝑙exp[𝑟^𝑢−1]. (2.2)

Lemma 2.2 (see [29]). Assume that {𝑥(𝑛)} satisfies 𝑥(𝑛+1)≥𝑥(𝑛)exp[𝑟(𝑛)−𝑎(𝑛)𝑥(𝑛)],𝑘≥𝑁₀, (2.3)limsup_𝑛→+∞𝑥(𝑛)≤𝑥^∗ and 𝑥(𝑁₀)>0, where {𝑟(𝑛)} and {𝑎(𝑛)} are all positive sequences bounded above and below by positive constants and 𝑁₀∈𝑁. Then liminf_𝑛→+∞𝑟𝑥(𝑛)≥^𝑙exp[𝑟^𝑙−𝑎^𝑢𝑥^∗]𝑎^𝑢. (2.4)

Lemma 2.3 (see [43]). Let 𝑥∶𝑍→𝑅 be a nonnegative bounded sequences, and let 𝐻∶𝑁→𝑅 be a nonnegative sequences such that ∑^∞_𝑛=0𝐻(𝑛)=1. Then liminf_𝑛→+∞𝑥(𝑛)≤liminf^𝑛_{𝑛→+∞𝑠=−∞}𝐻(𝑛−𝑠)𝑥(𝑠)≤limsup^𝑛_{𝑛→+∞𝑠=−∞}𝐻(𝑛−𝑠)𝑥(𝑠)≤limsup_𝑛→+∞𝑥(𝑛). (2.5)

3. Main Results

Now, we investigate the persistence property and stability property of system (1.1).

Theorem 3.1. Assume that 𝑎^𝑙₁−𝑐^𝑢₁𝑀₃>0,( (𝐻₁)) 𝑎^𝑙₂−𝑐^𝑢₂𝑀₃>0( (𝐻₂)) hold, then system (1.1) is permanent, that is, 𝑚_𝑖≤liminf_𝑛→+∞𝑥_𝑖(𝑛)≤limsup_𝑛→+∞𝑥_𝑖(𝑛)≤𝑀_𝑖,𝑖=1,2,3, (3.1) where 𝑀_𝑖=1𝑏^𝑙_𝑖exp[𝑎^𝑢_𝑖−1],𝑖=1,2,𝑀₃=1𝑏^𝑙₃exp[𝑑^𝑢₁𝑀₁+𝑑^𝑢₂𝑀₂𝑚−1],_𝑖=𝑎^𝑙_𝑖−𝑐^𝑢_𝑖𝑀₃𝑏^𝑢_𝑖exp[𝑎^𝑙_𝑖−𝑐^𝑢_𝑖𝑀₃−𝑏^𝑢_𝑖𝑀_𝑖𝑚],𝑖=1,2,₃=𝑑^𝑙₁𝑚₁+𝑑^𝑙₂𝑚₂𝑏₃exp[𝑑^𝑙₁𝑚₁+𝑑^𝑙₂𝑚₂−𝑏^𝑢₃𝑀₃]. (3.2)

Proof. It follows from the first two equations of system (1.1) that 𝑥_𝑖(𝑛+1)≤𝑥_𝑖(𝑛)exp[𝑎_𝑖(𝑛)−𝑏_𝑖(𝑛)𝑥_𝑖(𝑛)],𝑖=1,2. (3.3) So, as a consequence of Lemma 2.1, for any solution (𝑥₁(𝑛),𝑥₂(𝑛),𝑥₃(𝑛))^∞_𝑛=0 of system (1.1) with initial condition (1.3), one has limsup_𝑛→+∞𝑥_𝑖1(𝑛)≤𝑏^𝑙_𝑖exp[𝑎^𝑢_𝑖−1]=^def𝑀_𝑖,𝑖=1,2. (3.4) According to Lemma 2.3, from (3.4), we have limsup^𝑛_{𝑛→+∞𝑠=−∞}𝐻₃(𝑛−𝑠)𝑥₁(𝑠)≤𝑀₁,limsup^𝑛_{𝑛→+∞𝑠=−∞}𝐻₄(𝑛−𝑠)𝑥₂(𝑠)≤𝑀₂. (3.5) For any small positive constant 𝜀>0, it follows from (3.5) that there exists a positive integer 𝑛₁ such that for all 𝑛≥𝑛₁, ^𝑛_𝑠=−∞𝐻₃(𝑛−𝑠)𝑥₁(𝑠)≤𝑀₁+𝜀,^𝑛_𝑠=−∞𝐻₄(𝑛−𝑠)𝑥₂(𝑠)≤𝑀₂+𝜀. (3.6) Thus, for all 𝑛≥𝑛₁, from the last equation of system (1.1), if follows that 𝑥₃(𝑛+1)≤𝑥₃(𝑛)exp[−𝑏^𝑙₃𝑥₃(𝑛)+𝑑^𝑢₁(𝑀₁+𝜀)+𝑑^𝑢₂(𝑀₂+𝜀)]. (3.7) By applying Lemma 2.1 to (3.7), we have limsup_𝑛→+∞𝑥₃1(𝑛)≤𝑏^𝑙₃exp[𝑑^𝑢₁(𝑀₁+𝜀)+𝑑^𝑢₂(𝑀₂+𝜀)−1]. (3.8) Setting 𝜀→0, it follows that limsup_𝑛→+∞𝑥₃1(𝑛)≤𝑏^𝑙₃exp[𝑑^𝑢₁𝑀₁+𝑑^𝑢₂𝑀₂−1]=^def𝑀₃. (3.9) Next, we show that under the assumption of Theorem 3.1, liminf_𝑛→+∞𝑥_𝑖(𝑛)≥𝑚_𝑖,𝑖=1,2,3. (3.10) According to Lemma 2.3, from (3.9) we have limsup^𝑛_{𝑛→∞𝑠=−∞}𝐻_𝑖(𝑛−𝑠)𝑥₃(𝑠)≤𝑀₃,𝑖=1,2. (3.11) Conditions (𝐻₁) and (𝐻₂) imply that for enough small positive constant 𝜀₁, we have 𝑎^𝑙_𝑖−𝐶^𝑢_𝑖(𝑀₃+𝜀₁)>0,𝑖=1,2. (3.12) For 𝜀₁, it follows from (3.11) that there exists an positive integer 𝑛₂≥𝑛₁ such that for all 𝑛>𝑛₂,^𝑛_𝑠=−∞𝐻_𝑖(𝑛−𝑠)𝑥₃(𝑠)≤𝑀₃+𝜀₁,𝑖=1,2. (3.13) For 𝑛>𝑛₂, from (3.13) and the first two equations of system (1.1), we have 𝑥_𝑖(𝑛+1)≥𝑥_𝑖(𝑛)exp[𝑎^𝑙_𝑖−𝑏^𝑢_𝑖𝑥_𝑖(𝑛)−𝑐^𝑢_𝑖(𝑀₃+𝜀₁)],𝑖=1,2. (3.14) Thus, according to Lemma 2.2, one has liminf_𝑛→+∞𝑥_𝑖𝑎(𝑛)≥^𝑙𝑖−𝑐^𝑢_𝑖(𝑀₃+𝜀₁)𝑏^𝑢_𝑖exp[𝑎^𝑙_𝑖−𝑐^𝑢_𝑖(𝑀₃+𝜀₁)−𝑏^𝑢_𝑖𝑀_𝑖],𝑖=1,2. (3.15) Setting 𝜀₁→0, it follows that liminf_𝑛→+∞𝑥_𝑖𝑎(𝑛)≥^𝑙𝑖−𝑐^𝑢_𝑖𝑀₃𝑏^𝑢_𝑖exp[𝑎^𝑙_𝑖−𝑐^𝑢_𝑖𝑀₃−𝑏^𝑢_𝑖𝑀_𝑖]=^def𝑚_𝑖,𝑖=1,2. (3.16) According to Lemma 2.3, from (3.16) we have liminf^𝑛_{𝑛→+∞𝑠=−∞}𝐻₃(𝑛−𝑠)𝑥₁(𝑠)≥𝑚₁,liminf^𝑛_{𝑛→+∞𝑠=−∞}𝐻₄(𝑛−𝑠)𝑥₂(𝑠)≥𝑚₂. (3.17) For any 𝜀₂>0 small enough, without loss of generality, we may assume that 𝜀₂<(1/2)min{𝑚₁,𝑚₂}. From (3.17), it follows that there exists a 𝑛₃, such that for all 𝑛≥𝑛₃,^𝑛_𝑠=−∞𝐻₃(𝑛−𝑠)𝑥₁(𝑠)≥𝑚₁−𝜀₂,^𝑛_𝑠=−∞𝐻₄(𝑛−𝑠)𝑥₂(𝑠)≥𝑚₂−𝜀₂. (3.18) For 𝑛≥𝑛₃, from (3.18) and the last equation of (1.1), we have 𝑥₃(𝑛+1)≥𝑥₃(𝑛)exp[−𝑏^𝑢₃𝑥₃(𝑛)+𝑑^𝑙₁(𝑚₁−𝜀₂)+𝑑^𝑙₂(𝑚₂−𝜀₂)]. (3.19) By applying Lemma 2.2 to (3.19), it follows that liminf_𝑛→+∞𝑥₃𝑑(𝑛)≥^𝑙1(𝑚₁−𝜀₂)+𝑑^𝑙₂(𝑚₂−𝜀₂)𝑏^𝑢₃×exp[𝑑^𝑙₁(𝑚₁−𝜀₂)+𝑑^𝑙₂(𝑚₂−𝜀₂)−𝑏^𝑢₃𝑀₃]. (3.20) Setting 𝜀₂→0, it follows that liminf_𝑛→+∞𝑥₃𝑑(𝑛)≥^𝑙1𝑚₁+𝑑^𝑙₂𝑚₂𝑏^𝑢₃exp[𝑑^𝑙₁𝑚₁+𝑑^𝑙₂𝑚₂−𝑏^𝑢₃𝑀₃]=^def𝑚₃. (3.21) This ends the proof of Theorem 3.1.

Theorem 3.2. Assume that (𝐻₁) and (𝐻₂) hold. Assume further that there exist positive constants 𝛼,𝛽,𝛾 and 𝛿 such that 𝛼min{𝑏^𝑙₁,2𝑀₁−𝑏^𝑢₁}−𝛾𝑑^𝑢₁>𝛿,( (𝐻₃)) 𝛽min{𝑏^𝑙₂,2𝑀₂−𝑏^𝑢₂}−𝛾𝑑^𝑢₂>𝛿,( (𝐻₄)) 𝛾min{𝑏^𝑙₃,2𝑀₃−𝑏^𝑢₃}−𝛼𝑐^𝑢₁−𝛽𝑐^𝑢₂>𝛿,( (𝐻₅)) hold. Then for any two positive solutions (𝑥₁(𝑛),𝑥₂(𝑛),𝑥₃(𝑛))^𝑇 and (𝑦₁(𝑛),𝑦₂(𝑛),𝑦₃(𝑛))^𝑇 of system (1.1), one has lim_𝑛→+∞(𝑥_𝑖(𝑛)−𝑦_𝑖(𝑛))=0,𝑖=1,2,3. (3.22)

Proof. From conditions (𝐻₃)–(𝐻₅), there exits an enough small positive constant 𝜀₃ such that 𝛼min{𝑏^𝑙₁,2𝑀₁+𝜀₃−𝑏^𝑢₁}−𝛾𝑑^𝑢₁>𝛿, (3.23)𝛽min{𝑏^𝑙₂,2𝑀₂+𝜀₃−𝑏^𝑢₂}−𝛾𝑑^𝑢₂>𝛿, (3.24)𝛾min{𝑏^𝑙₃,2𝑀₃+𝜀₃−𝑏^𝑢₃}−𝛼𝑐^𝑢₁−𝛽𝑐^𝑢₂>𝛿. (3.25) Since (𝐻₁) and (𝐻₂) hold, for any solutions (𝑥₁(𝑛),𝑥₂(𝑛),𝑥₃(𝑛))^𝑇 and (𝑦₁(𝑛),𝑦₂(𝑛),𝑦₃(𝑛))^𝑇 of system (1.1) with the initial conditions (1.3), it follows from Theorem 3.1 that limsup_𝑛→+∞𝑥_𝑖(𝑛)≤𝑀_𝑖,limsup_𝑛→+∞𝑦_𝑖(𝑛)≤𝑀_𝑖,𝑖=1,2,3. (3.26) For the above 𝜀₃ and (3.26), there exists an 𝑛^∗>0 such that for all 𝑛>𝑛^∗, 𝑥_𝑖(𝑛)≤𝑀_𝑖+𝜀₃,𝑦_𝑖(𝑛)≤𝑀_𝑖+𝜀₃. (3.27) Firstly, let 𝑉₁₁(𝑛)=|ln𝑥₁(𝑛)−ln𝑦₁(𝑛)|. (3.28) Then from the first equation of the system (1.1), we have 𝑉₁₁(𝑛+1)=|ln𝑥₁(𝑛+1)−ln𝑦₁(𝑛+1)|≤|ln𝑥₁(𝑛)−ln𝑦₁(𝑛)−𝑏₁(𝑛)(𝑥₁(𝑛)−𝑦₁(𝑛))|+𝑐₁(𝑛)|^𝑛_𝑠=−∞𝐻₁(𝑛−𝑠)∣𝑥₃(𝑠)−𝑦₃(𝑠)|. (3.29) Using the Mean Value Theorem, we get 𝑥₁(𝑛)−𝑦₁(𝑛)=exp(ln𝑥₁(𝑛))−exp(ln𝑦₁(𝑛))=𝜉₁(𝑛)(ln𝑥₁(𝑛)−ln𝑦₁(𝑛)), (3.30) where 𝜉₁(𝑛) lies between 𝑥₁(𝑛) and 𝑦₁(𝑛), then it follows that 𝑉₁₁(𝑛+1)≤|ln𝑥₁(𝑛)−ln𝑦₁1(𝑛)|−(𝜉₁1(𝑛)−|𝜉₁(𝑛)−𝑏₁(𝑛)|)|𝑥₁(𝑛)−𝑦₁(𝑛)|+𝑐₁(𝑛)|^𝑛_𝑠=−∞𝐻₁(𝑛−𝑠)∣𝑥₃(𝑠)−𝑦₃(𝑠)|, (3.31) and so Δ𝑉₁₁1≤−(𝜉₁1(𝑛)−|𝜉₁(𝑛)−𝑏₁(𝑛)|)|𝑥₁(𝑛)−𝑦₁(𝑛)|+𝑐₁(𝑛)|^𝑛_𝑠=−∞𝐻₁(𝑛−𝑠)∣𝑥₃(𝑠)−𝑦₃(𝑠)|. (3.32) Secondly, let 𝑉₁₂(𝑛)=^∞_𝑠=0𝐻₁(𝑠)^𝑛−1_{𝜃=𝑛−𝑠}𝑐₁(𝜃+𝑠)|𝑥₃(𝜃)−𝑦₃(𝜃)|. (3.33) then, similar to the aforementioned analysis, we have Δ𝑉₁₂=𝑉₁₂(𝑛+1)−𝑉₁₂(𝑛)≤𝑐^𝑢₁|𝑥₃(𝑛)−𝑦₃(𝑛)|−𝑐₁(𝑛)^𝑛_𝑠=−∞𝐻₁(𝑛−𝑠)|𝑥₃(𝑠)−𝑦₃(𝑠)|. (3.34) Now, set 𝑉₁(𝑛)=𝑉₁₁(𝑛)+𝑉₁₂(𝑛), (3.35) then from (3.32) and (3.34), we have Δ𝑉₁=Δ𝑉₁₁+Δ𝑉₁₂1≤−(𝜉₁1(𝑛)−|𝜉₁(𝑛)−𝑏₁(𝑛)|)|𝑥₁(𝑛)−𝑦₁(𝑛)|+𝑐^𝑢₁|𝑥₃(𝑛)−𝑦₃(𝑛)|. (3.36) Let 𝑉₂(𝑛)=𝑉₂₁(𝑛)+𝑉₂₂(𝑛), (3.37) where 𝑉₂₁(𝑛)=|ln𝑥₂(𝑛)−ln𝑦₂𝑉(𝑛)|,₂₂(𝑛)=^∞_𝑠=0𝐻₂(𝑠)^𝑛−1_{𝜃=𝑛−𝑠}𝑐₂(𝜃+𝑠)|𝑥₃(𝜃)−𝑦₃(𝜃)|. (3.38) Similar to the aforementioned analysis, we have Δ𝑉₂=Δ𝑉₂₁+Δ𝑉₂₂1≤−(𝜉₂1(𝑛)−|𝜉₂(𝑛)−𝑏₂(𝑛)|)|𝑥₂(𝑛)−𝑦₂(𝑛)|+𝑐^𝑢₂|𝑥₃(𝑛)−𝑦₃(𝑛)|, (3.39) where 𝜉₂(𝑛) lies between 𝑥₂(𝑛) and 𝑦₂(𝑛).
Let
𝑉₃(𝑛)=𝑉₃₁(𝑛)+𝑉₃₂(𝑛), (3.40) where 𝑉₃₁(𝑛)=|ln𝑥₃(𝑛)−ln𝑦₃𝑉(𝑛)|,₃₂(𝑛)=^∞_𝑠=0𝐻₃(𝑠)^𝑛−1_{𝜃=𝑛−𝑠}𝑑₁(𝜃+𝑠)|𝑥₁(𝜃)−𝑦₁+(𝜃)|_∞𝑠=0𝐻₄(𝑠)^𝑛−1_{𝜃=𝑛−𝑠}𝑑₂(𝜃+𝑠)|𝑥₂(𝜃)−𝑦₂(𝜃)|. (3.41) Similar to the aforementioned analysis, we have Δ𝑉₃=Δ𝑉₃₁+Δ𝑉₃₂1≤−(𝜉₃1(𝑛)−|𝜉₃(𝑛)−𝑏₃(𝑛)|)|𝑥₃(𝑛)−𝑦₃(𝑛)|+𝑑^𝑢₁|𝑥₁(𝑛)−𝑦₁(𝑛)|+𝑑^𝑢₂|𝑥₂(𝑛)−𝑦₂(𝑛)|, (3.42) where 𝜉₃(𝑛) lies between 𝑥₃(𝑛) and 𝑦₃(𝑛).
Now, we define a Lyapunou functional as follows:
𝑉(𝑛)=𝛼𝑉₁(𝑛)+𝛽𝑉₂(𝑛)+𝛾𝑉₃(𝑛). (3.43) Calculating the difference of 𝑉 along the solution of system (1.1), for 𝑛>𝑛^∗, it follows from (3.23), (3.24), (3.25), (3.27), (3.36), (3.39) and (3.42) that Δ𝑉≤−[𝛼min{𝑏^𝑙₁,2𝑀₁+𝜀₃−𝑏^𝑢₁}−𝛾𝑑^𝑢₁]|𝑥₁(𝑛)−𝑦₁(𝑛)|−[𝛽min{𝑏^𝑙₂,2𝑀₂+𝜀₃−𝑏^𝑢₂}−𝛾𝑑^𝑢₂]|𝑥₂(𝑛)−𝑦₂(𝑛)|−[𝛾min{𝑏^𝑙₃,2𝑀₃+𝜀₃−𝑏^𝑢₃}−𝛼𝑐^𝑢₁−𝛽𝑐^𝑢₂]|𝑥₃(𝑛)−𝑦₃(𝑛)|≤−𝛿^𝑛_𝑖=1|𝑥_𝑖(𝑛)−𝑦_𝑖(𝑛)|. (3.44) Summating both sides of the above inequalities from 𝑛^∗ to 𝑛, we have ^𝑛_𝑝=𝑛∗(𝑉(𝑝+1)−𝑉(𝑝))≤−𝛿^𝑛_𝑝=𝑛∗³_𝑖=1|𝑥_𝑖(𝑝)−𝑦_𝑖(𝑝)|, (3.45) which implies 𝑉(𝑛+1)+𝛿^𝑛_𝑝=𝑛∗³_𝑖=1|𝑥_𝑖(𝑝)−𝑦_𝑖(𝑝)|≤𝑉(𝑛^∗). (3.46) It follows that ^𝑛_𝑝=𝑛∗³_𝑖=1|𝑥_𝑖(𝑝)−𝑦_𝑖(𝑝)|≤𝑉(𝑛^∗)𝛿, (3.47) then ^𝑛_𝑝=𝑛∗³_𝑖=1|𝑥_𝑖(𝑝)−𝑦_𝑖(𝑝)|≤𝑉(𝑛^∗)𝛿<+∞, (3.48) which means that lim_𝑛→∞∑³_𝑖=1|𝑥_𝑖(𝑛)−𝑦_𝑖(𝑛)|=0, that is lim_𝑛→+∞(𝑥_𝑖(𝑛)−𝑦_𝑖(𝑛))=0,𝑖=1,2,3. (3.49) This completes the proof of Theorem 3.2.

4. Example

The following example shows the feasibility of the main results.

Example 4.1. Consider the following system: 𝑥₁(𝑛+1)=𝑥₁(𝑛)exp[(1.1+0.1sin(√2𝑛))−(1.2+0.2sin(√2𝑛))𝑥₁(𝑛)−(0.3+0.2cos(𝑛))^𝑛_𝑠=−∞𝑒−1𝑒𝑒^{−(𝑛−𝑠)}𝑥₃𝑥(𝑠)],₂(𝑛+1)=𝑥₂(𝑛)exp[(1+0.1sin(√3𝑛))−(1.1+0.1sin(𝑛))𝑥₂√(𝑛)−(0.2+0.02cos(3𝑛))^𝑛_𝑠=−∞𝑒²−1𝑒²𝑒^{−2(𝑛−𝑠)}𝑥₃𝑥(𝑠)],₃(𝑛+1)=𝑥₃(𝑛)exp[−(1.15+0.15sin(√3𝑛))𝑥₃√(𝑛)+(0.12+0.03cos(2𝑛))^𝑛_𝑠=−∞𝑒³−1𝑒³𝑒^{−3(𝑛−𝑠)}𝑥₁(𝑠)+(0.13+0.03cos(𝑛))^𝑛_𝑠=−∞𝑒⁴−1𝑒⁴𝑒^{−4(𝑛−𝑠)}𝑥₂(𝑠)]. (4.1) One could easily see that there exist 𝛼=0.06,𝛽=0.05,𝛾=0.05 and 𝛿=0.001 such that 𝑎^𝑙₁−𝑐^𝑢₁𝑀₃𝑎=1−0.5𝑀−3≈0.7364>0,_𝑙2−𝑐^𝑢₂𝑀₃=1−0.22𝑀₃≈0.8840>0,𝛼min{𝑏^𝑙₁,2𝑀₁−𝑏^𝑢₁}−𝛾𝑑^𝑢₁≈0.0082>𝛿,𝛽min{𝑏^𝑙₂,2𝑀₂−𝑏^𝑢₂}−𝛾𝑑^𝑢₂≈0.0241>𝛿,𝛾min{𝑏^𝑙₃,2𝑀₃−𝑏^𝑢₃}−𝛼𝑐^𝑢₁−𝛽𝑐^𝑢₂≈0.009>𝛿. (4.2) Clearly, conditions (𝐻₁)–(𝐻₅) are satisfied. From Theorems 3.1 and 3.2, (1.1) is permanent and globally attractive. Numeric simulation (Figure 1) strongly supports our results.

Figure 1 

Dynamics behaviors of system (4.1) with initial conditions $(x_1(s),x_2(s),x_3(s){)}^T=(\mathrm{0.5,1},0.3{)}^T,(\mathrm{1.1,1.1,0.1}{)}^T,(\mathrm{0.9,0.6,0.2}{)}^T\text{,}$$s=\dots ,-n,-n+1,\dots ,-\mathrm{1,0}.$

5. Discussion

In this paper, we propose a discrete two-prey one-predator model with infinite delay. Theorem 3.1 shows that to ensure the permanence of the system, one should ensure 𝑎^𝑙_𝑖,𝑖=1,2 and 𝑏^𝑙₃ enough large, that is, the net birth rate of prey species and the density restriction of predator species should be large enough. We also obtain a set of sufficient conditions which ensures the global attractivity of the system.

Acknowledgments

The authors are grateful to anonymous referees for their excellent suggestions, which greatly improve the presentation of the paper. Also, this work was supported by the Program for New Century Excellent Talents in Fujian Province University (0330-003383).